Complex dynamics induced by strong confinement – From tracer diffusion in strongly heterogeneous media to glassy relaxation of dense fluids in narrow slits

We provide an overview of recent advances of the complex dynamics of particles in strong confinements. The first paradigm is the Lorentz model where tracers explore a quenched disordered host structure. Such systems naturally occur as limiting cases of binary glass-forming systems if the dynamics of one component is much faster than the other. For a certain critical density of the host structure the tracers undergo a localization transition which constitutes a critical phenomenon. A series of predictions in the vicinity of the transition have been elaborated and tested versus computer simulations. Analytical progress is achieved for small obstacle densities. The second paradigm is a dense strongly interacting liquid confined to a narrow slab. Then the glass transition depends nonmonotonically on the separation of the plates due to an interplay of local packing and layering. Very small slab widths allow to address certain features of the statics and dynamics analytically.     

Potential energy landscape and long-time dynamics in a simple model glass

We analyze the properties of a Lennard-Jones system at the level of the potential energy landscape. After an exhaustive investigation of the topological features of the landscape of the systems, obtained by studying small size samples, we describe the dynamics of the systems in multidimensional configurational space by means of a simple model. This considers the configurational space as a connected network of minima where the dynamics proceeds by jumps described by an appropriate master equation. Using this model we are able to reproduce the long-time dynamics and the low temperature regime. We investigate both the equilibrium regime and the off-equilibrium one, finding those typical glassy behaviors usually observed in the experiments such as (i) a stretched exponential relaxation, (ii) a temperature-dependent stretching parameter, (iii) a breakdown of the Stokes-Einstein relation, and (iv) the appearance of a critical temperature below which one observes a deviation from the fluctuation-dissipation relation as a consequence of the lack of equilibrium in the system.     

A Diffusive Strategic Dynamics for Social Systems

We propose a model for the dynamics of a social system, which includes diffusive effects and a biased rule for spin-flips, reproducing the effect of strategic choices. This model is able to mimic some phenomena taking place during marketing or political campaigns. Using a cost function based on the Ising model defined on the typical quenched interaction environments for social systems (Erdös-Renyi graph, small-world and scale-free networks), we find, by numerical simulations, that a stable stationary state is reached, and we compare the final state to the one obtained with standard dynamics, by means of total magnetization and magnetic susceptibility. Our results show that the diffusive strategic dynamics features a critical interaction parameter strictly lower than the standard one. We discuss the relevance of our findings in social systems.     

Koopman analysis of nonlinear systems with a neural network representation

The observation and study of nonlinear dynamical systems has been gaining popularity over years in different fields. The intrinsic complexity of their dynamics defies many existing tools based on individual orbits, while the Koopman operator governs evolution of functions defined in phase space and is thus focused on ensembles of orbits, which provides an alternative approach to investigate global features of system dynamics prescribed by spectral properties of the operator. However, it is difficult to identify and represent the most relevant eigenfunctions in practice. Here, combined with the Koopman analysis, a neural network is designed to achieve the reconstruction and evolution of complex dynamical systems. By invoking the error minimization, a fundamental set of Koopman eigenfunctions are derived, which may reproduce the input dynamics through a nonlinear transformation provided by the neural network. The corresponding eigenvalues are also directly extracted by the specific evolutionary structure built in.     

Dynamic interacting bubble simulation (DIBS): an agent-based bubble model for reacting fluidized beds

In this paper we explore the global dynamics of an agent-type model for bubbles in gas-fluidized beds and demonstrate that these features are consistent with experimentally observed behavior. The model accounts for the simultaneous interactions of thousands of individual bubbles and includes mass-transfer and first-order reactions between the gas and solids so that the impact of the dynamics is reflected in reactant conversion. We start with model parameters that have been demonstrated to produce time average behavior consistent with experimental reactor measurements. By observing the temporal variations of spatially averaged bubble properties, we are able to clearly distinguish the onset of global low-dimensional features that appear to be consistent with previous observations. The most prominent of these features is a large-scale oscillation that exhibits intermittency with power-law scaling in the vicinity of a critical gas flow. We show that the oscillation occurs as the result of a globally synchronized horizontal movement of the bubbles toward the center of the reactor. The oscillation appears to be consistent with the occurrence of the so-called "slugging" phenomenon, which is known to have large effects on fluidized bed reactor performance. Although this model can clearly be further improved, its success in replicating several of the key features of slugging indicates that this approach can serve as a useful tool for understanding and possibly controlling fluidized bed dynamics. We also conjecture that this model may be useful for more generally understanding the occurrence of global features in high-dimensional, multi-agent systems.     

Relaxation phenomena at criticality

The collective behaviour of statistical systems close to critical points is characterized by an extremely slow dynamics which, in the thermodynamic limit, eventually prevents them from relaxing to an equilibrium state after a change in the thermodynamic control parameters. The non-equilibrium evolution following this change displays some of the features typically observed in glassy materials, such as ageing, and it can be monitored via dynamic susceptibilities and correlation functions of the order parameter, the scaling behaviour of which is characterized by universal exponents, scaling functions, and amplitude ratios. This universality allows one to calculate these quantities in suitable simplified models and field-theoretical methods are a natural and viable approach for this analysis. In addition, if a statistical system is spatially confined, universal Casimir-like forces acting on the confining surfaces emerge and they build up in time when the temperature of the system is tuned to its critical value. We review here some of the theoretical results that have been obtained in recent years for universal quantities, such as the fluctuation-dissipation ratio, associated with the non-equilibrium critical dynamics, with particular focus on the Ising model with Glauber dynamics in the bulk. The non-equilibrium dynamics of the Casimir force acting in a film is discussed within the Gaussian model.     

Diffusive thermal dynamics for the spin-S Ising ferromagnet

We introduce an alternative thermal diffusive dynamics for the spin-S Ising ferromagnet realized by means of a random walker. The latter hops across the sites of the lattice and flips the relevant spins according to a probability depending on both the local magnetic arrangement and the temperature. The random walker, intended to model a diffusing excitation, interacts with the lattice so that it is biased towards those sites where it can achieve an energy gain. In order to adapt our algorithm to systems made up of arbitrary spins, some non trivial generalizations are implied. In particular, we will apply the new dynamics to two-dimensional spin-1/2 and spin-1 systems analyzing their relaxation and critical behavior. Some interesting differences with respect to canonical results are found; moreover, by comparing the outcomes from the examined cases, we will point out their main features, possibly extending the results to spin-S systems.     

Intermittent dynamics of critical fluctuations

We argue that the fluctuations of the order parameter in a complex system at the critical point can be described in terms of intermittent dynamics of type I. Based on this observation we develop an algorithm to calculate the isothermal critical exponent delta for a "thermal" critical system. We apply successfully our approach to the 3D Ising model. The intermittent character of these "critical" dynamics guides to the introduction of a new exponent which extends the notion of the exponent delta to nonthermal systems.     

Individual popularity and activity in online social systems

We propose a stochastic model of web user behaviors in online social systems, and study the influence of the attraction kernel on the statistical property of user or item occurrence. Combining the different growth patterns of new entities and attraction patterns of old ones, different heavy-tailed distributions for popularity and activity which have been observed in real life, can be obtained. From a broader perspective, we explore the underlying principle governing the statistical feature of individual popularity and activity in online social systems and point out the potential simple mechanism underlying the complex dynamics of the systems.     

Unraveling Hidden Major Factors by Breaking Heterogeneity into Homogeneous Parts within Many-System Problems

For a large ensemble of complex systems, a Many-System Problem (MSP) studies how heterogeneity constrains and hides structural mechanisms, and how to uncover and reveal hidden major factors from homogeneous parts. All member systems in an MSP share common governing principles of dynamics, but differ in idiosyncratic characteristics. A typical dynamic is found underlying response features with respect to covariate features of quantitative or qualitative data types. Neither all-system-as-one-whole nor individual system-specific functional structures are assumed in such response-vs-covariate (Re–Co) dynamics. We developed a computational protocol for identifying various collections of major factors of various orders underlying Re–Co dynamics. We first demonstrate the immanent effects of heterogeneity among member systems, which constrain compositions of major factors and even hide essential ones. Secondly, we show that fuller collections of major factors are discovered by breaking heterogeneity into many homogeneous parts. This process further realizes Anderson’s “More is Different” phenomenon. We employ the categorical nature of all features and develop a Categorical Exploratory Data Analysis (CEDA)-based major factor selection protocol. Information theoretical measurements—conditional mutual information and entropy—are heavily used in two selection criteria: C1—confirmable and C2—irreplaceable. All conditional entropies are evaluated through contingency tables with algorithmically computed reliability against the finite sample phenomenon. We study one artificially designed MSP and then two real collectives of Major League Baseball (MLB) pitching dynamics with 62 slider pitchers and 199 fastball pitchers, respectively. Finally, our MSP data analyzing techniques are applied to resolve a scientific issue related to the Rosenberg Self-Esteem Scale.     

Evolving complex food webs

We describe the properties of a model which links the ecology of food web structure with the evolutionary dynamics of speciation and extinction events; the model describes the dynamics of ecological communities on an evolutionary timescale. Species are defined as sets of characteristic features, and these features are used to determine interaction scores between species. A realistic population dynamics, which incorporates these scores, is used to determine the changes in population sizes on ecological time scales and so determine mean population sizes. We display typical examples of food webs constructed using the model and comment on the good agreement which is found between the model predictions and data on real webs.Received: 18 January 2004, Published online: 14 May 2004PACS: 87.23.Kg Dynamics of evolution - 87.23.Cc Population dynamics and ecological pattern formation - 89.75.Fb Structures and organization in complex systems     

Slow-fast dynamics in Josephson junctions

We report on a nontrivial type of slow-fast dynamics in a Josephson junction model externally shunted by a resistor and an inductor. For large values of the shunt inductance the slow manifold is highly folded, and different types of dynamical behavior in the fast variable are possible in dependence on the other parameters of the junction. We discuss how particular features of the dynamics are manifested in the current-voltage characteristics of the shunted junction.Received: 10 March 2003, Published online: 11 August 2003PACS: 05.45.-a Nonlinear dynamics and nonlinear dynamical systems - 85.25.Cp Josephson devices - 74.81.-g Inhomogeneous superconductors and superconducting systems     

Application of the ising model to the study of cluster multiplicities in finite excited systems

Fragmentation is expected to be an experimental signal which can give some information about the physical characteristics of finite excited systems. We try to investigate the question concerning the mechanism which governs the onset of the disassembly of such systems in connection with their internal dynamics. We restrict this study to the case where the systems are in thermodynamical equilibrium. We use the 2-dimensional Ising model as a generic model in order to calculate the multiplicities for the formation of strongly bound clusters inside the system and analyse their behaviour as a function of the temperature. The calculated observables show features which reflect the role played by the internal dynamics. The evolution of the shape of the cluster multiplicities with temperature obtained in the present study is qualitatively the same as the one which comes out through different nuclear phase space fragmentation models.     

Phase transitions and universality in nonequilibrium steady states of stochastic Ising models

We present results of direct computer simulations and of Monte Carlo renormalization group (MCRG) studies of the nonequilibrium steady states of a spin system with competing dynamics and of the voter model. The MCRG method, previously used only for equilibrium systems, appears to give useful information also for these nonequilibrium systems. The critical exponents are found to be of Ising type for the competing dynamics model at its second-order phase transitions, and of mean-field type for the voter model (consistent with known results for the latter).     

Topological aspects of a long-range model

The mean field XY Hamiltonian, a suitable model for studying long-range interactions in extended systems, presents, amongst other interesting features, slow relaxation and formation of quasi-stationary (QS) states. It is now known that, along these QS trajectories, the system visits critical points (maxima) of the potential energy function, characterized by a large number of directions with marginal stability. This observation may provide an interpretation for the slow relaxation dynamics and the trapping in such trajectories. In this paper we present further results and discussion on topological aspects of the model.     

Late stage, non-equilibrium dynamics in the dipolar Ising model

Magnetic domain structures are a fascinating area of study with interest deriving both from technological applications and fundamental scientific questions. The nature of the striped magnetic phases observed in ultra-thin films is one such intriguing system. The non-equilibrium dynamics of such systems as they evolve toward equilibrium has only recently become an area of interest and previous work on model systems showed evidence of complex, slow dynamics with glass-like properties as the stripes order mesoscopically. To aid in the characterization of the observed phases and the nature of the transitions observed in model systems we have developed an efficient method for identifying clusters or domains in the spin system, where the clusters are based on the stripe orientation. Thus we are able to track the growth and decay of such clusters of stripes in a Monte Carlo simulation and observe directly the nature of the slow dynamics. We have applied this method to consider the growth and decay of ordered domains after a quench from a saturated magnetic state to temperatures near and well below the critical temperature in the 2D dipolar Ising model. We discuss our method of identifying stripe domains or clusters of stripes within this model and present the results of our investigations.     

Noise-induced dynamical transition in systems with symmetric absorbing states

We investigate the effect of noise strength on the macroscopic ordering dynamics of systems with symmetric absorbing states. Using an explicit stochastic microscopic model, we present evidence for a phase transition in the coarsening dynamics, from an Ising-like to a voter-like behavior, as the noise strength is increased past a nontrivial critical value. By mapping to a thermal diffusion process, we argue that the transition arises due to locally-absorbing states being entered more readily in the high-noise regime, which in turn prevents surface tension from driving the ordering process.     

Noisy nonlinear coupled equations—Some new insights

We discuss the use of coupled nonlinear stochastic differential equations to model the dynamics of complex systems, and present some analytical insights into their critical behaviour. These concern in particular the role of infrared divergences which show up in a self-consistent resummation of perturbation theory (mode-coupling approximation), and their effects on critical exponents obtained in earlier work.     

Self organization of interacting polya urns

We introduce a simple model which shows non-trivial self organized critical properties. The model describes a system of interacting units, modelled by Polya urns, subject to perturbations and which occasionally break down. Three equivalent formulations - stochastic, quenched and deterministic - are shown to reproduce the same dynamics. Among the novel features of the model are a non-homogeneous stationary state, the presence of a non-stationary critical phase and non-trivial exponents even in mean field. We discuss simple interpretations in term of biological evolution and earthquake dynamics and we report on extensive numerical simulations in dimensions d=1,2 as well as in the random neighbors limit. Received: 18 February 1998 / Revised: 20 March 1998 / Accepted: 29 March 1998     

On the interplay between mathematics and biology: Hallmarks toward a new systems biology

This paper proposes a critical analysis of the existing literature on mathematical tools developed toward systems biology approaches and, out of this overview, develops a new approach whose main features can be briefly summarized as follows: derivation of mathematical structures suitable to capture the complexity of biological, hence living, systems, modeling, by appropriate mathematical tools, Darwinian type dynamics, namely mutations followed by selection and evolution. Moreover, multiscale methods to move from genes to cells, and from cells to tissue are analyzed in view of a new systems biology approach.